In another thread, the idea of a gunslinger variant designed around making a sling cool or useful came up; perhaps we could attempt it here? The way I see it these things need to happen to make it work.
Minimum things that need replacing or reworking:
Gunsmith - replace with something
Remove the touch attack aspect.
Find a replacement for gun training(s)
Remove or rework deadeye into something different.
Remove pistol-whip - perhaps re-flavor to allow the slinger to use the loaded sling like a sap up close?
Remove/replace and rework utility shot.
Remove expert loading
Rework lightning reload.
Add deeds specifically for slinger.
Increase crit modifier?
Increase damage done or rate of fire?
Ancient peoples used the sling in combat—armies included both specialist slingers and regular soldiers equipped with slings. As a weapon, the sling had several advantages; a sling bullet lobbed in a high trajectory can achieve ranges approaching 400 m; Modern authorities vary widely in their estimates of the effective range of ancient weapons. A bow and arrow could also have been used to produce long-range arcing trajectory, but ancient writers repeatedly stress the sling's advantage of range. The sling was light to carry and cheap to produce; ammunition in the form of stones was readily available and often to be found near the site of battle. The ranges the sling could achieve with molded lead glandes was only topped by the strong composite bow or, centuries later, the heavy English longbow, both at massively greater cost.
You will want to include further accuracy (and perhaps damage) bonuses at higher levels. You can take the easy route and just give the slingmeister "weapon training" as the fighter class feature to replace gun training. Will you be able to add dex AND strength to sling damage? If so then that is plenty enough damage and you could just focus on accuracy bonuses. I sort of think of adding dex to damage (from gun training) to be the sort of signature ability of the gunslinger class.